In a large saucepan over medium, melt the butter. Add the shallots and ½ teaspoon salt, then cook, stirring, until beginning to brown, about 5 minutes. Add the bulgur and cook, stirring constantly, until it has a nutty aroma, about 1 minute. Stir in 2¼ cups water and bring to a simmer over medium-high. Cover, reduce to medium-low and cook until the bulgur is tender and has absorbed the liquid, 15 to 17 minutes. Remove from the heat, uncover and drape a kitchen towel over the pan. Replace the cover and let stand for 5 minutes.